Nash-Williams theorem

In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and more generally forests) a graph can have:

A graph G has t edge-disjoint spanning trees iff for every partition ${\textstyle V_{1},\ldots ,V_{k}\subset V(G)}$ where ${\displaystyle V_{i}\neq \emptyset }$ there are at least t(k − 1) crossing edges.

The theorem was proved independently by Tutte^[1] and Nash-Williams,^[2] both in 1961. In 2012, Kaiser^[3] gave a short elementary proof.

For this article, we say that such a graph has arboricity t or is t-arboric. (The actual definition of arboricity is slightly different and applies to forests rather than trees.)

Related tree-packing properties

A k-arboric graph is necessarily k-edge connected. The converse is not true.

As a corollary of the Nash-Williams theorem, every 2k-edge connected graph is k-arboric.

Both Nash-Williams' theorem and Menger's theorem characterize when a graph has k edge-disjoint paths between two vertices.

Nash-Williams theorem for forests

In 1964, Nash-Williams^[4] generalized the above result to forests:

A graph ${\displaystyle G}$ can be partitioned into ${\displaystyle t}$ edge-disjoint forests iff for every ${\displaystyle U\subset V(G)}$, the induced subgraph ${\displaystyle G[U]}$ has at most ${\displaystyle t(|U|-1)}$ edges.

Other proofs are given here.^[5][6]

This is how people usually define what it means for a graph to be t-arboric.

In other words, for every subgraph ${\displaystyle S=G[U]}$, we have ${\displaystyle t\geq \lceil E(S)/(V(S)-1)\rceil }$. It is tight in that there is a subgraph ${\displaystyle S}$ that saturates the inequality (or else we can choose a smaller ${\displaystyle t}$). This leads to the following formula

${\displaystyle t=\lceil \max _{S\subset G}{\frac {E(S)}{V(S)-1}}\rceil }$,

also referred to as the Nash-Williams formula.

The general problem is to ask when a graph can be covered by edge-disjoint subgraphs.

See also

• Arboricity
• Bridge (cut edge)
• Matroid partitioning
• Menger's theorem
• Tree packing conjecture